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O-minimality
A structure (M,<,\ldots) is said to be o-minimal if every subset X \subset M^1 definable with parameters from M can be written as a finite union of points and intervals, i.e., as a boolean combination of sets of the form \{x \in M : x \le a\} and \{x \in M : x \ge a\} . Note that this is an assertion about subsets of M^1 , not definable sets in higher dimensions. This notion is analogous to minimality. In minimality, one assumes that the definable (one-dimensional) sets are quantifier-free definable using nothing but equality. Here, one assumes that the (one-dimensional) sets are quantifier-free definable using nothing but the ordering. A theory T with a predicate < is said to be o-minimal if every model of T is o-minimal. Unlike the case of minimality vs. strong minimality, there is no notion of strong o-minimality. It turns out that any elementary extension of an o-minimal structure is o-minimal. Consequently, the true theory of any o-minimal structure is an o-minimal theory. The proof of this is rather non-trivial, and uses the cell decomposition result for o-minimal theories. Often one restricts to the class of o-minimal structures/theories in which the ordering (M,<) is dense, i.e., a model of DLO. Most o-minimal theories of interest have this property, and many proofs can be simplified with this assumption. Examples Some relatively elementary examples: * DLO, the theory of dense linear orders. This is the true theory of (\mathbb{Q},<) . * RCF, the theory of real closed fields. This is the true theory of \mathbb{R} as an ordered field. * DOAG or ODAG, the theory of divisible ordered abelian groups. This is the true theory of (\mathbb{R},+,<) . By hard theorems of Alex Wilkie and other people, certain expansions of the ordered field \mathbb{R} are known to be o-minimal. * The structure \mathbb{R}_{exp} := (\mathbb{R},+,\cdot,\exp) was proven to be o-minimal by Alex Wilkie. This structure consists of the ordered field \mathbb{R} expanded by adding in a predicate for the exponentiation map. This example is somewhat surprising, given that we lack a recursive axiomatization of this structure. * The structure \mathbb{R}_{an} , consisting of the ordered field \mathbb{R} with restricted analytic functions, is o-minimal. For each real-analytic function f on an open neighborhood of 0,1^n , one adds a function symbol for f restricted to 0,1 . This does not subsume \mathbb{R}_{exp} , since \exp turns out to not be definable in \mathbb{R}_{an} . In fact the o-minimality of \mathbb{R}_{an} is a more basic result. It is essentially Gabrielov's theorem. * More generally, \mathbb{R}_{an,exp} is o-minimal. This is the expansion of \mathbb{R} obtained by adding in both the exponential map and the restricted analytic functions. * More generally, one can add all Pfaffian functions. The most general result in this direction is due to Speissegger, maybe. Properties O-minimal theories are NIP, but never stable or simple, as they have the order property. O-minimal theories are also superrosy, of finite rank. In any o-minimal theory, definable closure and algebraic closure agree (on account of the ordering), and these operations define a pregeometry on the home sort. This yields a well-defined notion of dimension of definable sets. Not all o-minimal theories eliminate imaginaries, even after naming all parameters from a model. However, o-minimal expansions of RCF always eliminate imaginaries, and in fact have definable choice (which includes definable Skolem functions). The same holds for o-minimal expansions of DOAG after naming at least one non-zero element. Definable functions and definable sets have many nice structural properties. For simplicity assume that the order is dense. Then one has the following results: * Every definable function f : M^1 \to M^n is piecewise continuous: the domain of f can be written as a finite union of intervals, such that on each interval, f is continuous. If n = 1 , then one can also arrange that on each interval, f is either constant, or strictly increasing, or strictly decreasing. * Every definable subset of M^n has finitely many definably connected components. In the presence of definable Skolem functions, each piece is definably path-connected. * More precisely, every definable subset X \subset M^n has a cell-decomposition: it can be written as disjoint union of sets that are "cells" in a certain sense. Each cell is definably connected, and in the case of o-minimal expansions of RCF, is definably homeomorphic to a ball. * If f : M^n \to M^m is a definable function, then the domain of f has a cell decomposition such that the restriction of f to each cell is continuous. * If X \subset M^n , the topological closure \overline{X} of X has dimension no bigger than X , and the frontier \overline{X} \setminus X has strictly lower dimension than X . Many of the topological pathologies that are common in pointset topology and real analysis don't occur when working with definable sets in o-minimal expansions of the reals. For example, every definable set is locally path-connected, every connected component is path connected, every set without interior is nowhere dense, and every definable set is homotopy equivalent to a finite simplicial complex. Moreover, every continuous definable function is piecewise differentiable, and in fact piecewise C^k for every k < \infty . One also knows that if f and g are two definable functions \mathbb{R} \to \mathbb{R} , then f and g are asymptotically comparable. Limits always exist (possibly taking values \pm \infty ). These results apply in particular to, e.g., the structure (\mathbb{R},+,\cdot,\exp) . In sharp contrast, the definable sets in (\mathbb{R},+,\cdot,\sin) are exactly the sets in the projective hierarchy, so e.g. there are definable sets which are not Borel. O-minimal trichotomy Some analog of the Zilber trichotomy holds in the o-minimal setting. Applications Real algebraic geometry, Pila-Wilkie…